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Discrete Math and Analyzing Social Graphs
Week 2
Sums of Binomial Coefficients

QuizQuiz • 15 min15 minutes

Sums of Binomial Coefficients

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Sums of Binomial Coefficients

Graded Quiz • 15 min

Due Mar 2, 2:59 AM EST

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Sums of Binomial Coefficients

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1.

Question 1

What is the sum of all the elements in the sixth rows of Pascal's triangle?

1 / 1 point

32

Correct

Exactly! Indeed, the sum of binomial coefficients in the $i$ -th row is equal to $2^{i}$

2.

Question 2

What is the sum of all the elements from the first six rows of Pascal's triangle?

1 / 1 point

63

Correct

Exactly! It is $1+2+4+8+16+32=2^1+2^2+2^3+2^4+2^5=2^6-1=63$ .

3.

Question 3

What is the value of the sum $\binom{5}{3}+\binom{5}{4}+\binom{5}{5}$ ?

1 / 1 point

16

Correct

Exactly! We can compute the sum explicitly, but we can also observe that for all $i$ we have $\binom{5}{i} = \binom{5}{5-i}$ . So, the sum of these three coefficients is exactly half of the sum of all coefficients in the sixth row.